Syllabus
Basic concepts of probability theory: classical and axiomatic definition of probability, conditional probability, independence of random events, the law of total probability, Bayes' theorem.
Random variables: Definition of a random variable, its distribution and its distribution function, their properties, discrete and continuous distributions, mean value and variance of a random variable, other numerical characteristics of random variables, distribution of functions of random variables.
Random vectors: Definition of a random vector, its distribution and its distribution function, independence of random variables, numerical characteristics of random vectors, distribution of functions of random vectors.
Conditional distribution and conditional expectation. Transformation of random variables and random vectors. Characteristic function and moment generating function.
Stochastic inequalities: Chebyshev's inequality, Markov's inequality, Hoeffding's inequality, Mill's inequality, Cauchy-Schwartz inequality, Jensen's inequality.
Stochastic convergence: Convergence in probability, convergence in distribution, convergence in L2.
Limit theorems: Weak law of large numbers, central limit theorem, delta method.
Statistics: Foundations and basic concepts of statistics, random sample.
Parametric models: Point and interval estimation. Unbiased, consistent estimates. Method of moments, maximum likelihood method. Overview of basic interval estimation (based on normality and CLV).
Hypothesis testing: Formulation of statistical hypotheses, type I error, type II error, significance level, p-value.
Empirical distribution function. Statistical functionals. Bootstrap.
Annotation
An introductory course in probability theory and statistics. Required course for General Mathematics.